Most railroad track can be divided into alternating sections of straight track and of curved track. Each section of curved track can in turn be divided into sections in which the curvature is constant throughout the section and sections in which the curvature varies with distance along the section. In a section of straight track the bank angle of the track is normally zero (with a possible exception near either end of the section). In a section of curved track that has constant curvature and that is not restricted to very low train speed the bank angle is normally greater than zero and constant (again with a possible exception near either end of the section).
Between a section of straight track with zero bank angle and a section of curved track with curvature and bank angle constant and non-zero it is necessary to have a transition section in which bank angle varies with distance so as to match the adjacent bank angle at each end. Normally the curvature of such a transition section also varies with distance and matches the curvature of the adjacent section at each end. Such a transition is referred to as a spiral. In the original and most widely used spiral the bank angle and curvature both vary linearly with distance along the transition section. A spiral in which the curvature varies linearly with distance has an alignment shape referred to in the railroad industry as a clothoid spiral.
The bank angle of the track will be generally referred to hereinafter as the “roll angle”. The roll angle of the track will determine and be the same as the roll angle of a vehicle wheel set about the longitudinal axis (i.e., the axis that is in the plane of the track and that is parallel to the local direction of the track). Roll in the sense of banking as used herein should not be confused with roll in the sense that a vehicle wheel rolls about an axis that is in the plane of the track but approximately perpendicular to the local direction of the track.
The description of this invention will refer to the curvature of the track. The curvature of the track is a property of the alignment of the track as seen in plan view. It is equal to the derivative of the local compass direction of the track (in radians) with respect to distance along the track. The curvature at a point on the track is also equal to the reciprocal of the radius of a circle for which the derivative of the compass bearing along the periphery with respect to peripheral distance is the same as that of the spiral.
The description of this invention will refer to the “offset” between two neighboring sections of track, each of which has constant curvature. The offset between two such adjacent track sections is the smallest distance between extensions of the sections that maintain their respective fixed curvatures. The offset can be assumed to be greater than zero and must be so in order for adjacent constant curvature sections to be connected by a spiral with monotonically varying curvature.
It has long been recognized that when a rail vehicle travels over a clothoid spiral the vehicle is subjected to abrupt lateral and roll accelerations that cause a little discomfort to passengers and whose reaction forces on the track structure degrade the alignment of the track. As a result, a number of alternate forms of variation of spiral curvature with distance have been proposed, some of which have been used in practice. Alternate methods for design of railroad transition spirals that have been proposed and used in the past are described in Bjorn Kufver, VTI Report 420A, “Mathematical Description of Railway Alignments and Some Preliminary Comparative Studies”, Swedish National Road and Transport Research Institute (1997).
In addition, there has been consideration of the height of the roll axis, which is the longitudinal axis about which the track is rotated for purpose of changing the roll angle. It has been proposed and proven in practice that spiral performance can be substantially improved if the roll axis is raised above the plain of the track. This technique is described in Gerard Presle and Herbert L. Hasslinger, “Entwicklung und Grundlagen neuer Gleisgeometrie”, ZEV+DET Glas. Ann. 122, 1998, 9/10, September/October, page 579.
All of the previously published methods for design of a railroad spiral begin by specifying a functional form for the curvature of the track as a function of distance along the spiral. Also, to the extent presently known, all of the previously published formulae for curvature of spirals lead to a discontinuity in the third derivative of the track curvature at each end of the spiral.